Alternate derivation of Ginocchio-Haxton relation [(2j+3)/6]

We address the problem, previously considered by Ginocchio and Haxton (GH), of the number of states for three identical particles in a single j shell with angular momentum J=j. GH solved this problem in the context of the quantum Hall effect. We address it in a more direct way. We also consider the case J=j+1 to demonstrate that our method is more general, and we show how to take care of added complications for a system of five identical particles